Question

A normal distribution has a mean of 142 and a standard deviation of 16. What is the probability that a randomly selected value lies between 142 and 174? A) 0.22 B) 0.34 C) 0.48 D) 0.54

Answer 1

`\mathbb P(142<X<174)=\mathbb P\left((142-142)/(16)<(X-142)/(16)<(174-142)/(16)\right)=\mathbb P(0<Z<2)`

Approximately 95% of any normal distribution lies within two standard deviations of the mean, i.e.
`\mathbb P(-2<Z<2)\approx0.95`. Because the distribution is symmetric, you have
`\mathbb P(-2<Z<2)=2\mathbb P(0<Z<2)`, so
`\mathbb P(0<Z<2)\approx\frac{0.95}2=0.475\approx0.48`.